Synaptic

To get a list of packages available in a PPA / repository enabled on your system (if the repository is disabled or not added on your system, it won't show up here), firstly install Synaptic if you haven't already:

Then open Synaptic, select "Origin" on the bottom left, then select a PPA or repository on the left and it should list all packages in that PPA / repository for your Ubuntu version, both installed and not installed:

In the latest Synaptic built with GTK3, there are two entries for each PPA here, one that uses "/now" at the end, which displays the packages you've installed from a repository and another one which displays all the packages available in that repository.

Important: if the exact same package (including the package version) exists in two or more PPAs / repositories, it will only be listed for one repository entry in Synaptic. That's why I've added other ways of doing this (see below), which list all the packages, even if the package exists in other PPAs / repositories.

Command line

You can also do this using a command like the one below, but this will only list the package names, without any additional info (no version, description, etc.):

Using Y PPA Manager

Y PPA Manager, a small application I've created, can list all the packages available in each PPA added on your system. It doesn't work with regular repositories (just Launchpad PPAs) though and the PPA needs to be enabled on your system for this to work. So it's not perfect.

But there are advantages over Synaptic / the command above:

unlike Synaptic, it lists all the packages in a PPA, even if the exact same package (exact version) is available in multiple PPAs;

unlike the command above, Y PPA Manager displays the package version next to the package name.